Exercise EE.C11
Find the characteristic polynomial of the matrix \(A=\left[ \begin{matrix} 3 & 2 & 1 \\ 0 & 1 & 1 \\ 1 & 2 & 0 \end{matrix} \right]\)
We first find the determinant of \(\left( A\quad -\quad \lambda \times I \right)\). In the videos from Khan Academy we’ve seen that they substract matrix A from lambda times the identity matrix, but we arrive at the same result either way.
Thus,
\(\left| A\quad -\quad \lambda \times I \right| =\left| \left[ \begin{matrix} 3 & 2 & 1 \\ 0 & 1 & 1 \\ 1 & 2 & 0 \end{matrix} \right] -\left[ \begin{matrix} \lambda & 0 & 0 \\ 0 & \lambda & 0 \\ 0 & 0 & \lambda \end{matrix} \right] \right| =0\)
\(\left| \left[ \begin{matrix} 3-\lambda & 2 & 1 \\ 0 & 1-\lambda & 1 \\ 1 & 2 & 0-\lambda \end{matrix} \right] \right| =0\)
\(3-\lambda \left| \begin{matrix} 1-\lambda & 1 \\ 2 & 0-\lambda \end{matrix} \right| -\quad 0\left| \begin{matrix} 2 & 1 \\ 2 & 0-\lambda \end{matrix} \right| +1\left| \begin{matrix} 2 & 1 \\ 1-\lambda & 1 \end{matrix} \right| =0\)
\(3-\lambda \left[ (1-\lambda )(0-\lambda )-2 \right] -0+1\left[ 2-(1-\lambda \right] =0\)
\(3-\lambda (-\lambda +{ \lambda }^{ 2 }-2)+2-1+\lambda =0\)
\(-3\lambda +3{ \lambda }^{ 2 }-6+{ \lambda }^{ 2 }-{ \lambda }^{ 3 }+2\lambda +2-1+\lambda =0\)
The characteristic polynomial is:
\(-{ \lambda }^{ 3 }+4{ \lambda }^{ 2 }-5\)